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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">c'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>, <font color="Maroon">d'</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:45_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>,<span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>,<span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">fg</font> = <font color="Maroon">x</font> as    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>,<span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<a NAME="E3:45_1_1"/><i><font color="Green">E19</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E5:45_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> )
 <b>and </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x1</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="cat_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="cat_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:45_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 
<b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, <a class="txt" href="cat_2.html#E5:45_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E10:45_1_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> )
 
<b>by </b><i><a class="ref" href="cat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:45_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font> )
 
;<br/>
<b>hence </b><a NAME="E12:45_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font> )
 <b>and </b><br/><a NAME="E5:45_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x1</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">c'</font> <b>by </b><i><a class="txt" href="cat_2.html#E5:45_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x2</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">d</font>,<font color="Maroon">d'</font> <b>by </b><i><a class="txt" href="cat_2.html#E5:45_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><a NAME="E8:45_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E5:45_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E9:45_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>,<span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="cat_2.html#K12" target="_self">]</a></span>,<span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="cat_2.html#K12" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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